Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{2k^3 + 4k^2 - 70k}{4k^3 - 52k^2 + 160k}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {2k(k^2 + 2k - 35)} {4k(k^2 - 13k + 40)} $ $ p = \dfrac{2k}{4k} \cdot \dfrac{k^2 + 2k - 35}{k^2 - 13k + 40} $ Simplify: $ p = \dfrac{1}{2} \cdot \dfrac{k^2 + 2k - 35}{k^2 - 13k + 40}$ Since we are dividing by $k$ , we must remember that $k \neq 0$ Next factor the numerator and denominator. $ p = \dfrac{1}{2} \cdot \dfrac{(k - 5)(k + 7)}{(k - 5)(k - 8)}$ Assuming $k \neq 5$ , we can cancel the $k - 5$ $ p = \dfrac{1}{2} \cdot \dfrac{k + 7}{k - 8}$ Therefore: $ p = \dfrac{ k + 7 }{ 2(k - 8)}$, $k \neq 5$, $k \neq 0$